Essential dimension of finite groups in prime characteristic

Let F be a field of characteristic $p>0$ and G be a smooth finite algebraic group over F. We compute the essential dimension ${\mathrm{ed}}_F(G;p)$ of G at p. That is, we show that

${\mathrm{ed}}_F(G;p)=\{\begin{array}{cc}1,\hfill & \text{if}p\text{divides}|G|,\text{and}\hfill \\ 0,\hfill & \text{otherwise.}\hfill \end{array}$